Dic₇×C₇₃ #14523284

order	2044
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₄×C₇₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₇×C₇₃
unambiguous semidirect product	C₇⋊(C₄×C₇₃)

Splittings

derived, Sylow	C₇ ⋊ C₄×C₇₃
Hall	C₇×C₇₃ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇₃	D₇
commutator, hypocenter	C₇	C₄×C₇₃
Frattini	C₂	D₇×C₇₃
Fitting, socle, prime radical	C₂×C₇×C₇₃	C₂
cosocle	C₂×C₇	C₂×C₇₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₇₃

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 2044

A minimal faithful permutation representation — 2 generators on 84 points:

(1,2)(3,4)(5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84);(1,3,2,4)(6,11)(7,10)(8,9)

Conjugacy classes & character table

730 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇₃×Dic₄₉, C₇×Dic₇×C₇₃, C₇₃×C₇⋊Dic₇, …
maximal normal subgroup of	C₇×Dic₇×C₇₃, C₁₁×Dic₇×C₇₃, C₁₃×Dic₇×C₇₃, …
maximal quotient of	C₇₃×Dic₄₉, C₇×Dic₇×C₇₃, C₇₃×C₇⋊Dic₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2044, 2);

Dic7×C73 #14523284